{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"7de635cc9832744aa19554bd26cc2f95d93b5f701aadd4d74206ab0d7ed842cf","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5114783ff66bd8f125a9f87f228a25f155ebd2e1e05e40216615d0eeb5d83988","5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"b29491442d205b6a550c9b7e42a2623801dffed2f2f809675a7fe5a8623c4c36","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.ionization.no-dianions","kind":"well-posed","literature_status":"open","n":"1","parents":[{"note":"","parent_revision":"5114783ff66bd8f125a9f87f228a25f155ebd2e1e05e40216615d0eeb5d83988","relation":"special_case"}],"plain":"Experiments find no stable atom carrying two extra electrons. Prove from the Schrodinger equation that $N_c(Z) \\le Z + 1$ for every element.","posed_since":"","precise":"With $H_{N,Z}$ and $N_c(Z)$ as in mathph.ionization.excess-charge-bound (static nucleus, nonrelativistic, spin $1/2$ electrons), prove $N_c(Z) \\le Z + 1$ for all $Z \\ge 1$, or exhibit $Z$ with $N_c(Z) \\ge Z + 2$. A computer-assisted proof for small $Z$ combined with an asymptotic argument for large $Z$ would count.","problem_ref":null,"references":"","settled_by":"A proof that $E(N, Z) = E(N-1, Z)$ for every $N \\ge Z + 2$ and every $Z$, i.e. that no ion with two or more extra electrons has a bound ground state.","status_note":"Lieb's bound $N_c < 2Z + 1$ gives this statement only for hydrogen ($Z = 1$); no stable doubly charged atomic negative ion has been observed.","title":"No atom binds two extra 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